Let α (a) and β (a) be the roots of the equation = 0, where a >

Engineering

Mathematics

Methods to Evaluate Limits

Question

Let α (a) and β (a) be the roots of the equation $(\sqrt[3]{1 + a} - 1) x^{2} + (\sqrt{1 + a} - 1) x + (\sqrt[6]{1 + a} - 1)$ = 0, where a > – 1. Then $\underset{a \to 0^{+}}{Lim} α (a) and \underset{a \to 0^{+}}{Lim} β (a)$ are

$\frac{- 9}{2} and 3$

$\frac{- 1}{2} and - 1$

$\frac{- 5}{2} and 1$

$\frac{- 7}{2} and 2$

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$(\sqrt[3]{1 + a} - 1) x^{2} + (\sqrt{1 + a} - 1) x + (\sqrt[6]{1 + a} - 1) = 0$

Let 1+ a = t⁶ ........(i)

when a → 0 + ⇒ t → 1

∴ Given equation becomes

(t² – 1) x² + (t³ – 1) x + (t – 1) = 0

(t + 1) x² + (t² + t + 1) x + 1 = 0 ⇒ 2x² + 3x + 1 = 0 ⇒ x = – 1 or $\frac{- 1}{2}$

$∴ \underset{a \to 0^{+}}{Lim} α (0) = - 1 and \underset{a \to 0^{+}}{Lim} β (0) = \frac{- 1}{2}$